Abstract
A Jacobi structure J on a line bundle \(L\rightarrow M\) is weakly regular, if the sharp map \(J^\sharp : J^1 L \rightarrow DL\) has constant rank. A generalized contact bundle with a weakly regular Jacobi structure possesses a transverse complex structure. Similarly to conditions obtained by Bailey on generalized complex structures, we find condition on a pair consisting of a weakly regular Jacobi structure and a transverse complex structure to come from a generalized contact structure. In this way, we are able to construct interesting examples of generalized contact bundles. As applications: (1) we prove that every five-dimensional nilmanifold is equipped with an invariant generalized contact structure; (2) we show that unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally, we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that does not possess a compatible generalized contact structure.
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Schnitzer, J. Weakly regular Jacobi structures and generalized contact bundles. Ann Glob Anal Geom 56, 221–244 (2019). https://doi.org/10.1007/s10455-019-09665-w
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DOI: https://doi.org/10.1007/s10455-019-09665-w